Some Discrete Methods for Boundary Integral Equations on Smooth Closed Curves

Convergence Results for Discrete Trigonometric Collocation Methods with Product Integration in Hölder-Zygmund Spaces

“…and then by (A.9) obtain 11 < C 11011i, IHIH-' with any a E R and i > max(Ia --i i, Ia -i i) + 2, verifying (2.7) in the case of A.…”

“…The basic trigonometric collocation method was discussed in [3], and fully discretized versions in [8] for operators of order 0 and in [1,4, 111 for operators of arbitrary order. For other full discretizations that have been presented for operators of some particular types see, e.g., the references of [11,121. The method of [8] is actually a quadrature method based on the use of integral representation of the operator and any Fourier representation is riot needed for applying the scheme.…”

“…For application of the methods of [1,4] the previously mentioned specific decomposition of the operator is essential since the main part is discretized based on its Fourier representation; the smooth perturbation is replaced by the trapezoidal rule approximation. The method of [11] is applicable to an operator of a form more general than it is the case in [1,3,4], but there also the explicit Fourier representation of the main part is needed. The analysis of the methods of [1,8,11,12] is given with respect to Sobolev norms, and the maximum norm error estimates of optimal order, which means the convergence of the same order as for trigonometric interpolation, are not achieved.…”

“…The method of [11] is applicable to an operator of a form more general than it is the case in [1,3,4], but there also the explicit Fourier representation of the main part is needed. The analysis of the methods of [1,8,11,12] is given with respect to Sobolev norms, and the maximum norm error estimates of optimal order, which means the convergence of the same order as for trigonometric interpolation, are not achieved. In [3, 4] Hölder-Zygmund norms are used, and pointwise error estimates of optimal order are derived for the methods involved in the case of operators of integer order.…”

“…Moreover, we present maximum norm error estimates in the case of boundary integral operators of integer order. Concerning the error analysis, we utilize the approach relying on the concepts of stability, consistency and convergence known from [6,8] and also from [9,10,11]. Our analysis is different from that of [3, 41 especially because of the consistency estimates, describing the accuracy of approximation when discretizing operators by using product integration.…”